Interleaving

Abstract

This chapter examines the interleaving relation between persistence modules, and the associated interleaving metric. Interleavings are approximate isomorphisms, and in the first instance may be defined by a pair of ‘shifted’ homomorphisms between the two persistence modules being compared. More abstractly, an interleaving can be thought of as a solution to a functor extension problem. The Interpolation Lemma is the main result of this chapter: it asserts that a pair of interleaved persistence modules can be interpolated by a 1-Lipschitz 1-parameter family. We give three different explicit constructions of the interpolation; two of them are the left and right Kan extensions (in the functor extension point of view), while the third mediates between the two.